Stat 5101 Lecture Notes - School of Statistics

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120 Stat 5101 (Geyer) Course Notesrandom variable taking values 0, 1, 2, .... If A is a bounded set and the pointprocess is boundedly finite, then the event N A = ∞ has probability zero.A point x is a fixed atom if P (N {x} > 0) > 0, that is, if there is positiveprobability of seeing a point at the particular location x in every random pattern.We are interested in point processes in which the locations of the points arecontinuous random variables, in which case the probability of seeing a point atany particular location is zero, so there are no fixed atoms.For a general spatial point process, the joint distribution of the variables N Afor various sets A is very complicated. There is one process for which it is notcomplicated. This is the Poisson process, which is a model for a “completelyrandom” pattern of points. One example of this process is given in Figure 4.1.Figure 4.1: A single realization of a homogeneous Poisson process.4.4.2 The Poisson ProcessA Poisson process is a spatial point process characterized by a simple independenceproperty.
4.4. THE POISSON PROCESS 121Definition 4.4.1.A Poisson process is a simple, boundedly finite spatial point process withno fixed atoms having the property that N A1 , N A2 , ..., N Ak are independentrandom variables, whenever A 1 , A 2 , ..., A k are disjoint bounded sets.In short, counts of points in disjoint regions are independent random variables.It is a remarkable fact that the independence property alone determinesthe distribution of the counts.Theorem 4.4. For a Poisson process, N A has a Poisson distribution for everybounded set A. Conversely, a simple point process with no fixed atoms such thatN A has a Poisson distribution for every bounded set A is a Poisson process.Write Λ(A) =E(N A ). Since the parameter of the Poisson distribution is themean, the theorem says N A has the Poisson distribution with parameter Λ(A).The function Λ(A) is called the intensity measure of the process.An important special case of the Poisson process occurs when the intensitymeasure is proportional to ordinary measure (length in one dimension, area intwo, volume in three, and so forth): if we denote the ordinary measure of aregion A by m(A), thenΛ(A) =λm(A) (4.9)for some λ>0. The parameter λ is called the rate parameter of the process. APoisson process for which (4.9) holds, the process is said to be a homogeneousPoisson process. Otherwise it is inhomogeneous.The space could be the three-dimensional space of our ordinary experience.For example, the points could be the locations of raisins in a carrot cake. Ifthe process is homogeneous, that models the situation where regions of equalvolume have an equal number of raisins on average, as would happen if thebatter was stirred well and the raisins didn’t settle to the bottom of the cakepan before baking. If the process is inhomogeneous, that models the situationwhere some regions get more raisins per unit volume than others on average.Either the batter wasn’t stirred well or the raisins settled or something of thesort.There are two important corollaries of the characterization theorem.Corollary 4.5. The sum of independent Poisson random variables is a Poissonrandom variable.If X i ∼ Poi(µ i ) then the X i could be the counts N Ai in disjoint regions A ihaving measures m(A i )=µ i in a homogeneous Poisson process with unit rateparameter. The sum is the count in the combined regionX 1 + ···+X n =N A1∪···∪A nwhich has a Poisson distribution with meanm(A 1 ∪···∪A n )=m(A 1 )+···m(A n )
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Stat 5101 Lecture NotesCharles J. G
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Contents1 Random Variables and Chan
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CONTENTSv5.1.4 Variance Matrices ..
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CONTENTSviiD Relations Among Brand
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Chapter 1Random Variables andChange
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1.1. RANDOM VARIABLES 3Sometimes it
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1.1. RANDOM VARIABLES 5that is, X i
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1.2. CHANGE OF VARIABLES 7the union
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1.2. CHANGE OF VARIABLES 9Thus even
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1.2. CHANGE OF VARIABLES 11Every in
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1.2. CHANGE OF VARIABLES 13where f
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1.3. RANDOM VECTORS 151.3.1 Discret
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1.4. THE SUPPORT OF A RANDOM VARIAB
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1.5. JOINT AND MARGINAL DISTRIBUTIO
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1.5. JOINT AND MARGINAL DISTRIBUTIO
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1.6. MULTIVARIABLE CHANGE OF VARIAB
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Chapter 2Expectation2.1 Introductio
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2.3. BASIC PROPERTIES 33expectation
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2.3. BASIC PROPERTIES 35that is, th
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2.4. MOMENTS 37The Multiplicativity
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2.4. MOMENTS 39holds for all real-v
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2.4. MOMENTS 41simple, it is often
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2.4. MOMENTS 43In contrast, for all
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2.4. MOMENTS 45is the sum of the ex
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2.4. MOMENTS 47Proof. Just take a i
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2.4. MOMENTS 49What happens to Coro
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2.4. MOMENTS 51This inequality is a
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2.4. MOMENTS 53(a)Why does (2.7) as
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2.5. PROBABILITY THEORY AS LINEAR A
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Page 77 and 78: 2.5. PROBABILITY THEORY AS LINEAR A
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Page 85 and 86: 2.7. INDEPENDENCE 772.7 Independenc
Page 87 and 88: 2.7. INDEPENDENCE 79Then X and Y ar
Page 89 and 90: 2.7. INDEPENDENCE 81Show that the f
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Page 119 and 120: Chapter 4Parametric Families ofDist
Page 121 and 122: 4.1. LOCATION-SCALE FAMILIES 113The
Page 123 and 124: 4.2. THE GAMMA DISTRIBUTION 115Show
Page 125 and 126: 4.3. THE BETA DISTRIBUTION 117cours
Page 127: 4.4. THE POISSON PROCESS 119Hence t
Page 131 and 132: 4.4. THE POISSON PROCESS 123measure
Page 133 and 134: 4.4. THE POISSON PROCESS 1254-3. A
Page 135 and 136: Chapter 5Multivariate DistributionT
Page 137 and 138: 5.1. RANDOM VECTORS 129Again like r
Page 139 and 140: 5.1. RANDOM VECTORS 1315.1.6 Covari
Page 141 and 142: 5.1. RANDOM VECTORS 1335.1.7 Linear
Page 143 and 144: 5.1. RANDOM VECTORS 135Example 5.1.
Page 145 and 146: 5.1. RANDOM VECTORS 137Example 5.1.
Page 147 and 148: 5.1. RANDOM VECTORS 139where we hav
Page 149 and 150: 5.2. THE MULTIVARIATE NORMAL DISTRI
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Page 161 and 162: 5.3. BERNOULLI RANDOM VECTORS 153yo
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Page 173 and 174: Chapter 6Convergence Concepts6.1 Un
Page 175 and 176: 6.1. UNIVARIATE THEORY 167It simpli
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6.1. UNIVARIATE THEORY 171Rewriting
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6.1. UNIVARIATE THEORY 173Comment T
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6.1. UNIVARIATE THEORY 175Problems6
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Chapter 7Sampling Theory7.1 Empiric
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7.1. EMPIRICAL DISTRIBUTIONS 1797.1
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7.1. EMPIRICAL DISTRIBUTIONS 181Def
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7.1. EMPIRICAL DISTRIBUTIONS 183To
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7.2. SAMPLES AND POPULATIONS 185The
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7.2. SAMPLES AND POPULATIONS 187•
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7.3. SAMPLING DISTRIBUTIONS OF SAMP
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Appendix AGreek LettersTable A.1: T
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Appendix BSummary of Brand-NameDist
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B.1. DISCRETE DISTRIBUTIONS 215B.1.
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B.2. CONTINUOUS DISTRIBUTIONS 217Th
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B.2. CONTINUOUS DISTRIBUTIONS 219B.
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B.4. DISCRETE MULTIVARIATE DISTRIBU
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B.5. CONTINUOUS MULTIVARIATE DISTRI
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B.5. CONTINUOUS MULTIVARIATE DISTRI
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Appendix CAddition Rules forDistrib
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Appendix DRelations Among BrandName
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Appendix EEigenvalues andEigenvecto
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E.2. EIGENVALUES AND EIGENVECTORS 2
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E.2. EIGENVALUES AND EIGENVECTORS 2
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E.3. POSITIVE DEFINITE MATRICES 237
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Appendix FNormal Approximations for
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